A lemma due to J.Nash [Ann.Math.(2) 56, 405–421 (1952; Zbl 0048.38501)] states that a manifold \(M\) analytically embedded in \(\mathbb R^n\) has a neighborhood \(N\) in \(\mathbb R^n\) in which there is a unique nearest point \(y \in M\) for each \(x \in N\), and in which \(y\) depends analytically on \(x\).
In the present paper this is generalized to definable/subanalytic sets \(M\) with parameters. More precisely: Let \(M \subseteq \mathbb R_t^k \times \mathbb R_x^n\) be nonempty with locally closed \(t\)-sections \(M_t\). Suppose that \(M\) is definable (in some fixed o-minimal structure). Then there exists a definable set \(W \subseteq \mathbb R_t^k \times \mathbb R_x^n\) with open \(t\)-sections \(W_t\) such that \(M_t \subseteq W_t\) is closed in \(W_t\) and \(m(t,x) \neq \emptyset\) for \(x \in W_t\), where \[ m(t,x):=\{y \in M_t : \|x-y\|=\text{dist}(x,M_t)\}, \quad (t,x) \in W. \] Moreover, the multifunction \(m(t,x)\) is definable; there is a definable set \(E \subseteq W\) with nowhere dense sections such that \(m(t,x)\) is single-valued iff \(x \in W_t \setminus E_t\); and for any integer \(p \geq 2\) there is a definable set \(E \subseteq F^p \subseteq W\) with closed and nowhere dense sections such that \(M_t\) is a \(C^p\)-submanifold near \(x \not\in F^p_t\) and \(m(t,\cdot)\) is \(C^{p-1}\) near \(x \in W_t \setminus \overline{E_t}\) iff \(x \not\in F^p_t\).
The author gives also a version for subanalytic sets \(M \subseteq \mathbb R^n\). (Note that the subanalytic subsets of \(\mathbb R^n\) do not form an o-minimal structure, in contrast to globally subanalytic sets.)
Finally, properties of \(m\) as a multifunction are investigated.